Average Error: 0.2 → 0.0
Time: 17.8s
Precision: 64
\[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]
↓
\[6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]
\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}
↓
6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}
double f(double x) {
        double r56490741 = 6.0;
        double r56490742 = x;
        double r56490743 = 1.0;
        double r56490744 = r56490742 - r56490743;
        double r56490745 = r56490741 * r56490744;
        double r56490746 = r56490742 + r56490743;
        double r56490747 = 4.0;
        double r56490748 = sqrt(r56490742);
        double r56490749 = r56490747 * r56490748;
        double r56490750 = r56490746 + r56490749;
        double r56490751 = r56490745 / r56490750;
        return r56490751;
}
↓
double f(double x) {
        double r56490752 = 6.0;
        double r56490753 = x;
        double r56490754 = 1.0;
        double r56490755 = r56490753 - r56490754;
        double r56490756 = sqrt(r56490753);
        double r56490757 = 4.0;
        double r56490758 = r56490753 + r56490754;
        double r56490759 = fma(r56490756, r56490757, r56490758);
        double r56490760 = r56490755 / r56490759;
        double r56490761 = r56490752 * r56490760;
        return r56490761;
}

Error

Bits error versus x

Target

Original0.2
Target0.1
Herbie0.0
\[\frac{6}{\frac{\left(x + 1\right) + 4 \cdot \sqrt{x}}{x - 1}}\]

Derivation

  1. Initial program 0.2

    \[\frac{6 \cdot \left(x - 1\right)}{\left(x + 1\right) + 4 \cdot \sqrt{x}}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{6}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}}\]
  3. Using strategy rm

  4. Applied div-inv0.1

    \[\leadsto \color{blue}{6 \cdot \frac{1}{\frac{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}{x - 1}}}\]

  5. Simplified0.0

    \[\leadsto 6 \cdot \color{blue}{\frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}}\]

  6. Final simplification0.0

    \[\leadsto 6 \cdot \frac{x - 1}{\mathsf{fma}\left(\sqrt{x}, 4, x + 1\right)}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "Data.Approximate.Numerics:blog from approximate-0.2.2.1"

  :herbie-target
  (/ 6.0 (/ (+ (+ x 1.0) (* 4.0 (sqrt x))) (- x 1.0)))

  (/ (* 6.0 (- x 1.0)) (+ (+ x 1.0) (* 4.0 (sqrt x)))))